people.math.sc.edu

1 Complex Variablesby R. B. Ash and NovingerPrefaceThis book represents a substantial revision of the first edition which was published in1971. Most of the topics of the original edition have been retained, but in a number ofinstances the material has been reworked so as to incorporate alternative approaches tothese topics that have appeared in the mathematical literature in recent book is intended as a text, appropriate for use by advanced undergraduates or gradu-ate students who have taken a course in introductory real analysis, or as it is often called,advanced calculus. No background in comple xvariables is assumed, thus making the te xtsuitable for those encountering the subject for the first time. It should be possible tocover the entire book in two list below enumerates many of the major changes and/or additions to the first The relationship between real-differentiability and the Cauchy-Riemann Dixon s proof of the homology version of Cauchy s The use of hexagons in tiling the plane, instead of squares, to characterize simpleconnectedness in terms of winding numbers of cycles.

2 This avoids troublesome detailsthat appear in the proofs where the tiling is done with Sandy Grabiner s simplified proof of Runge s A self-contained approach to the problem of extending Riemann maps of the unit diskto the boundary. In particular, no use is made of the Jordan curve theorem, a difficulttheorem which we believe to be peripheral to a course in comple xanalysis. Severalapplications of the result on extending maps are Newman s proof of the prime number theorem, as modified by J. Korevaar, ispresented in the last chapter as a means of collecting and applying many of the ideas andresults appearing in earlier chapters, while at the same time providing an introduction toseveral topics from analytic number the most part, each section is dependent on the previous ones, and we recommendthat the material be covered in the order in which it appears. Problem sets follow mostsections, with solutions provided (in a separate section).112We have attempted to provide careful and complete explanations of the material, whileat the same time maintaining a writing style which is succinct and to the Copyright 2004 by Ash and Novinger.

3 Paper or electronic copies for non-commercial use may be made freely without explicit permission of the authors. All otherrights are : 1 2 3 4 5 6 72 (Preface-2)TOC IndexComplex Variablesby Robert B. Ash and NovingerTable Of ContentsChapter 1: Basic Further Topology of the Analytic Real-Differentiability and the Cauchy-Riemann The Exponential Harmonic FunctionsChapter 2: The Elementary Integration on The Exponential Function and the Complex Trigonometric Further ApplicationsChapter 3: The General Cauchy Logarithms and The Index of a Point with Respect to a Closed Cauchy s Another Version of Cauchy s TheoremChapter 4: Applications of the Cauchy Residue The Open mapping Theorem for Analytic Linear Fractional Conformal Analytic Mappings of One Disk to Another1Ch: 1 2 3 4 5 6 73 (Table of Contents-1)TOC The Poisson Integral formula and its The Jensen and Poisson-Jensen Analytic ContinuationChapter 5: Families of Analytic The SpacesA( ) andC( ) The Riemann Mapping Extending Conformal Maps to the BoundaryChapter 6.

4 Factorization of Analytic Infinite Weierstrass Mittag-Leffler s Theorem and ApplicationsChapter 7: The Prime Number The Riemann Zeta An Equivalent Version of the Prime Number Proof of the Prime Number TheoremCh: 1 2 3 4 5 6 74 (Table of Contents-2)TOC IndexChapter 1 IntroductionThe reader is assumed to be familiar with the complex planeCto the extent found inmost college algebra texts, and to have had the equivalent of a standard introductorycourse in real analysis (advanced calculus). Such a course normally includes a discussionof continuity, differentiation, and Riemann-Stieltjes integration of functions from the realline to itself. In addition, there is usually an introductory study of metric spaces and theassociated ideas of open and closed sets, connectedness, convergence, compactness, andcontinuity of functions from one metric space to another. For the purpose of review andto establish notation, some of these concepts are discussed in the following Basic DefinitionsThe complex planeCis the set of all ordered pairs (a,b) of real numbers, with additionand multiplication defined by(a,b)+(c,d)=(a+c,b+d) and (a,b)(c,d)=(ac bd,ad+bc).

5 Ifi=(0,1) and the real numberais identified with (a,0), then (a,b)=a+bi. Theexpressiona+bican be manipulated as if it were an ordinary binomial expression of realnumbers, subject to the relationi2= 1. With the above definitions of addition andmultiplication,Cis a +bi, thenais called thereal partofz, writtena=Rez, andbis called theimaginary partofz, writtenb=Imz. Theabsolute valueormagnitudeormodulusofzis defined as (a2+b2)1/2. A complex number with magnitude 1 is said to (written argz) is defined as the angle which the line segment from (0,0)to (a,b) makes with the positive real axis. The argument is not unique, but is determinedup to a multiple of 2 .Ifris the magnitude ofzand is an argument ofz, we may writez=r(cos +isin )and it follows from trigonometric identities that|z1z2|=|z1||z2|and arg(z1z2) = argz1+ argz21Ch: 1 2 3 4 5 6 75 (1-1)TOC Index2 CHAPTER 1. INTRODUCTION(that is, if kis an argument ofzk,k=1,2, then 1+ 2is an argument ofz1z2). Ifz2 = 0, then arg(z1/z2) = arg(z1) arg(z2).

6 Ifz=a+bi, theconjugateofzis defined asz=a bi, and we have the following properties:|z|=|z|,argz= argz,z1+z2=z1+z2,z1 z2=z1 z2,z1z2=z1z2,Rez=(z+z)/2,Imz=(z z)/2i, zz=|z| two complex numbersz1andz2is defined asd(z1,z2)=|z1 z2|.Sod(z1,z2) is simply the Euclidean distance betweenz1andz2regarded as points inthe plane. Thusddefines a metric onC, and furthermore,dis complete, that is, everyCauchy sequence converges. Ifz1,z2,..is sequence of complex numbers, thenzn zifand only if Rezn Rezand Imzn Imz. We say thatzn if the sequence of realnumbers|zn|approaches + .Many of the above results are illustrated in the following analytical proof of the triangleinequality:|z1+z2| |z1|+|z2|for allz1,z2 geometric interpretation is that the length of a side of a triangle cannot exceed thesum of the lengths of the other two sides. See Figure , which illustrates the familiarrepresentation of complex numbers as vectors in the z1+z2 z2 Figure proof is as follows:|z1+z2|2=(z1+z2)(z1+z2)=|z1|2+|z 2|2+z1z2+z1z2=|z1|2+|z2|2+z1z2+z1z2=|z1| 2+|z2|2+ 2Re(z1z2) |z1|2+|z2|2+2|z1z2|=(|z1|+|z2|) proof is completed by taking the square root of both complex numbers, [a,b] denotes the closed line segment with arbitrary real numbers witht1<t2, then we may write[a,b]={a+t t1t2 t1(b a):t1 t t2}.

7 The notation is extended as follows. Ifa1,a2,..,an+1are points inC,apolygonfroma1toan+1(or a polygon joininga1toan+1) is defined asn j=1[aj,aj+1],often abbreviated as [a1,..,an+1].Ch: 1 2 3 4 5 6 76 (1-2)TOC FURTHER TOPOLOGY OF THE Further Topology of the PlaneRecall that two subsetsS1andS2of a metric space areseparatedif there are open setsG1 S1andG2 S2such thatG1 S2=G2 S1= , the empty set. A set isconnectediff it cannot be written as the union of two nonempty separated sets. An open(respectively closed) set is connected iff it is not the union of two nonempty disjoint open(respectively closed) DefinitionA setS Cis said to bepolygonally connectedif each pair of points inScan be joinedby a polygon that lies connectedness is a special case of path (or arcwise) connectedness, and itfollows that a polygonally connected set, in particular a polygon itself, is connected. Wewill prove in Theorem that anyopenconnected set is polygonally DefinitionsIfa Candr>0, thenD(a,r) is the open disk with centeraand radiusr;thusD(a,r)={z:|z a|<r}.

8 The closed disk{z:|z a| r}is denoted byD(a,r), andC(a,r) is the circle with centeraand TheoremIf is an open subset ofC, then is connected iff is polygonally If is connected anda , let 1be the set of allzin such that there is apolygon in fromatoz, and let 2= \ 1, there is an open diskD(z,r) (because is open). Ifw D(z,r), a polygon fromatozcan be extended tow, andit follows thatD(z,r) 1, proving that 1is open. Similarly, 2is open. (Supposez 2, and chooseD(z,r) . ThenD(z,r) 2as before.)Thus 1and 2are disjoint open sets, and 1 = becausea 1. Since isconnected we must have 2= , so that 1= . Therefore is polygonally converse assertion follows becauseanypolygonally connected set is connected. 4 DefinitionsAregioninCis an open connected subset ofC. A setE Cisconvexif for each pairof pointsa,b E, we have [a,b] E;Eisstarlikeif there is a pointa E(called astar center) such that [a,z] Efor eachz E. Note that any nonempty convex set isstarlike and that starlike sets are polygonally : 1 2 3 4 5 6 77 (1-3)TOC Index4 CHAPTER 1.

9 Analytic DefinitionLetf: C, where is a subset ofC. We say thatfiscomplex-differentiableat thepointz0 if for some Cwe havelimh 0f(z0+h) f(z0)h= (1)or equivalently,limz z0f(z) f(z0)z z0= .(2)Conditions (3), (4) and (5) below are also equivalent to (1), and are sometimes easier f(z0+hn) f(z0)hn= (3)for each sequence{hn}such thatz0+hn \{z0}andhn 0asn .limn f(zn) f(z0)zn z0= (4)for each sequence{zn}such thatzn \{z0}andzn z0asn .f(z)=f(z0)+(z z0)( + (z))(5)for allz , where : Cis continuous atz0and (z0)= show that (1) and (5) are equivalent, just note that may be written in terms offas follows: (z)= f(z) f(z0)z z0 ifz =z00ifz= number is unique. It is usually written asf (z0), and is called complex-differentiable at every point of ,fis said to beanalyticorholomorphicon . Analytic functions are the basic objects of study in complex on a nonopen setS Cmeans analyticity on an open set ,fis analytic at a pointz0ifffis analytic on an open set withz0.

10 Iff1andf2are analytic on , so aref1+f2,f1 f2,kf1fork C,f1f2, andf1/f2(providedthatf2is never 0 on ). Furthermore,(f1+f2) =f 1+f 2,(f1 f2) =f 1 f 2,(kf1) =kf 1(f1f2) =f1f 2+f 1f2, f1f2 =f2f 1 f1f : 1 2 3 4 5 6 78 (1-4)TOC REAL-DIFFERENTIABILITY AND THE CAUCHY-RIEMANN EQUATIONS5 The proofs are identical to the corresponding proofs for functions (z) = 1 by direct computation, we may use the rule for differentiating aproduct (just as in the real case) to obtainddz(zn)=nzn 1,n=0,1,..This extends ton= 1, 2,..using the quotient analytic on andgis analytic onf( ) ={f(z):z }, then the compositiong fis analytic on andddzg(f(z)) =g (f(z)f (z)just as in the real variable an example of the use of Condition (4) of ( ), we now prove a result that willbe useful later in studying certain inverse TheoremLetgbe analytic on the open set 1, and letfbe a continuous complex-valued functionon the open set . Assume(i)f( ) 1,(ii)g is never 0,(iii)g(f(z)) =zfor allz (thusfis 1-1).)